Bunce was born and raised in New Zealand, making his debut for Auckland in 1986. But he was first capped at international level for Western Samoa. The hard-hitting midfielder played four games for Western Samoa as they reached the quarter-finals of the 1991 Rugby World Cup in England.

Bunce's impact at international level brought him to the attention of All Blacks coach Laurie Mains, who selected him for the All Blacks in 1992. Bunce became the immovable man in the All Blacks backline, missing just one game (against Japan in the 1995 Rugby World Cup) before bowing out of the international arena against England in 1997.

The outside centre struck up a formidable partnership with Walter Little in the All Blacks' midfield, which reached a peak at the 1995 Rugby World Cup. The duo helped New Zealand to the final where they lost to South Africa. Bunce scored 20 tries in 55 games for the All Blacks.

From 2001 to 2003 he coached Italy before taking up a role with Auckland as defensive coach and then technical advisor to Samoa in 2005.